The word models \(\mathcal {W}\)  and \(\mathcal {W}^{\prime }\)  parallel the two best-studied random graph models on \(n\)  labeled vertices. For graphs, \(\mathcal {G}(n,p)\)  selects every edge with probability \(p\)  independently, and \(\mathcal {G}^{\prime }(n,m)\)  selects exactly \(m\)  edges uniformly from all possible ways [1]}. While the two kinds of models share many asymptotic properties, they differ in some important aspects, especially regarding subgraph counts, and in our models – subword counts.
